DHARMPROPOSITIONAL LOGIC 113· The formula (X ∨Y) can get value F under ‘v’ if and only if both X and Y get value F
under ‘v’.
· The formula (X → Y) can get value F under ‘v’ if and only if X get the value T and Y get
value F under ‘v’.
Boolean valuation provides the truth value to the formula over connectives: ~, ∧, ∨, →.
For a formula, construction of truth table is a two step method.
Step 1. Prepare all possible combinations of truth values for propositional variables
(that gives the total number of rows in the truth table). Number of variables gives the initial
number of columns.
Step 2. Obtain the truth values of each connective and put these truth values in a new
column.
Entries of the final column show the truth values of the formula.
Example 5.7. Construct the truth value for (P ∧ ~ Q)
Step 1. Since, formula (P ∧ ~ Q) has two variables (2 columns), so there are four possible
combinations of truth values for P and Q (4 rows).
Step 2. Add column 3 (for connective negation) to determine truth values of ~ Q, add
column 4 (for connective conjunction) to determine truth values of (P ∧ ~ Q).
Since there are no more connectives left, hence we get the truth table shown in Fig. 5.15
and the truth values of the formula (P ∧ ~ Q) are shown in column 4.
PQ~ QP ∧ ~ Q
FF T F
FT F F
TF T T
TT F F
12 3 4
Fig. 5.15
Example 5.8. Construct the truth table for the formula,
((P → Q) ∧ (~ P ∨ Q)) ∨ ( ~ ( P → Q) ∧ ( ~ P ∨ Q)) : (let it be X)
P Q (P → Q) ~P(~P ∨ Q) ~(P → Q) ~(~P ∨ Q) (P → Q) ∧ (~P ∨ Q) ~(P → Q) ∧ ~(~P ∨ Q) X
FFTTT F F T F T
FTTTT F F T F T
TFFFF T T F T T
TTTFT F F T F T
12345 6 7 8 9 10
Fig. 5.16